Properties

Label 3380.2.a.r.1.3
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 19 x^{7} + 16 x^{6} + 106 x^{5} - 87 x^{4} - 153 x^{3} + 149 x^{2} - 26 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.40883\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.40883 q^{3} -1.00000 q^{5} +2.17051 q^{7} -1.01520 q^{9} +O(q^{10})\) \(q-1.40883 q^{3} -1.00000 q^{5} +2.17051 q^{7} -1.01520 q^{9} +5.73367 q^{11} +1.40883 q^{15} -5.24468 q^{17} +7.00804 q^{19} -3.05787 q^{21} -8.19819 q^{23} +1.00000 q^{25} +5.65673 q^{27} +9.54822 q^{29} +2.41917 q^{31} -8.07777 q^{33} -2.17051 q^{35} -7.16094 q^{37} -7.16158 q^{41} +7.76780 q^{43} +1.01520 q^{45} +5.41944 q^{47} -2.28890 q^{49} +7.38886 q^{51} +3.94223 q^{53} -5.73367 q^{55} -9.87313 q^{57} +0.993988 q^{59} -4.28702 q^{61} -2.20350 q^{63} -11.6322 q^{67} +11.5499 q^{69} -5.19073 q^{71} +3.06083 q^{73} -1.40883 q^{75} +12.4450 q^{77} +7.33131 q^{79} -4.92377 q^{81} -2.86146 q^{83} +5.24468 q^{85} -13.4518 q^{87} +12.2808 q^{89} -3.40820 q^{93} -7.00804 q^{95} +7.35295 q^{97} -5.82083 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + O(q^{10}) \) \( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.40883 −0.813388 −0.406694 0.913564i \(-0.633318\pi\)
−0.406694 + 0.913564i \(0.633318\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.17051 0.820374 0.410187 0.912001i \(-0.365463\pi\)
0.410187 + 0.912001i \(0.365463\pi\)
\(8\) 0 0
\(9\) −1.01520 −0.338400
\(10\) 0 0
\(11\) 5.73367 1.72877 0.864384 0.502833i \(-0.167709\pi\)
0.864384 + 0.502833i \(0.167709\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 1.40883 0.363758
\(16\) 0 0
\(17\) −5.24468 −1.27202 −0.636011 0.771680i \(-0.719417\pi\)
−0.636011 + 0.771680i \(0.719417\pi\)
\(18\) 0 0
\(19\) 7.00804 1.60776 0.803878 0.594795i \(-0.202767\pi\)
0.803878 + 0.594795i \(0.202767\pi\)
\(20\) 0 0
\(21\) −3.05787 −0.667283
\(22\) 0 0
\(23\) −8.19819 −1.70944 −0.854721 0.519088i \(-0.826272\pi\)
−0.854721 + 0.519088i \(0.826272\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.65673 1.08864
\(28\) 0 0
\(29\) 9.54822 1.77306 0.886530 0.462672i \(-0.153109\pi\)
0.886530 + 0.462672i \(0.153109\pi\)
\(30\) 0 0
\(31\) 2.41917 0.434496 0.217248 0.976116i \(-0.430292\pi\)
0.217248 + 0.976116i \(0.430292\pi\)
\(32\) 0 0
\(33\) −8.07777 −1.40616
\(34\) 0 0
\(35\) −2.17051 −0.366883
\(36\) 0 0
\(37\) −7.16094 −1.17725 −0.588626 0.808406i \(-0.700331\pi\)
−0.588626 + 0.808406i \(0.700331\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.16158 −1.11845 −0.559225 0.829016i \(-0.688901\pi\)
−0.559225 + 0.829016i \(0.688901\pi\)
\(42\) 0 0
\(43\) 7.76780 1.18458 0.592290 0.805725i \(-0.298224\pi\)
0.592290 + 0.805725i \(0.298224\pi\)
\(44\) 0 0
\(45\) 1.01520 0.151337
\(46\) 0 0
\(47\) 5.41944 0.790506 0.395253 0.918572i \(-0.370657\pi\)
0.395253 + 0.918572i \(0.370657\pi\)
\(48\) 0 0
\(49\) −2.28890 −0.326986
\(50\) 0 0
\(51\) 7.38886 1.03465
\(52\) 0 0
\(53\) 3.94223 0.541508 0.270754 0.962649i \(-0.412727\pi\)
0.270754 + 0.962649i \(0.412727\pi\)
\(54\) 0 0
\(55\) −5.73367 −0.773128
\(56\) 0 0
\(57\) −9.87313 −1.30773
\(58\) 0 0
\(59\) 0.993988 0.129406 0.0647031 0.997905i \(-0.479390\pi\)
0.0647031 + 0.997905i \(0.479390\pi\)
\(60\) 0 0
\(61\) −4.28702 −0.548897 −0.274448 0.961602i \(-0.588495\pi\)
−0.274448 + 0.961602i \(0.588495\pi\)
\(62\) 0 0
\(63\) −2.20350 −0.277615
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.6322 −1.42110 −0.710548 0.703648i \(-0.751553\pi\)
−0.710548 + 0.703648i \(0.751553\pi\)
\(68\) 0 0
\(69\) 11.5499 1.39044
\(70\) 0 0
\(71\) −5.19073 −0.616026 −0.308013 0.951382i \(-0.599664\pi\)
−0.308013 + 0.951382i \(0.599664\pi\)
\(72\) 0 0
\(73\) 3.06083 0.358243 0.179121 0.983827i \(-0.442674\pi\)
0.179121 + 0.983827i \(0.442674\pi\)
\(74\) 0 0
\(75\) −1.40883 −0.162678
\(76\) 0 0
\(77\) 12.4450 1.41824
\(78\) 0 0
\(79\) 7.33131 0.824837 0.412419 0.910994i \(-0.364684\pi\)
0.412419 + 0.910994i \(0.364684\pi\)
\(80\) 0 0
\(81\) −4.92377 −0.547085
\(82\) 0 0
\(83\) −2.86146 −0.314086 −0.157043 0.987592i \(-0.550196\pi\)
−0.157043 + 0.987592i \(0.550196\pi\)
\(84\) 0 0
\(85\) 5.24468 0.568866
\(86\) 0 0
\(87\) −13.4518 −1.44219
\(88\) 0 0
\(89\) 12.2808 1.30176 0.650881 0.759180i \(-0.274400\pi\)
0.650881 + 0.759180i \(0.274400\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.40820 −0.353414
\(94\) 0 0
\(95\) −7.00804 −0.719010
\(96\) 0 0
\(97\) 7.35295 0.746579 0.373289 0.927715i \(-0.378230\pi\)
0.373289 + 0.927715i \(0.378230\pi\)
\(98\) 0 0
\(99\) −5.82083 −0.585015
\(100\) 0 0
\(101\) −13.4407 −1.33740 −0.668699 0.743533i \(-0.733148\pi\)
−0.668699 + 0.743533i \(0.733148\pi\)
\(102\) 0 0
\(103\) 7.68074 0.756806 0.378403 0.925641i \(-0.376473\pi\)
0.378403 + 0.925641i \(0.376473\pi\)
\(104\) 0 0
\(105\) 3.05787 0.298418
\(106\) 0 0
\(107\) −8.06583 −0.779754 −0.389877 0.920867i \(-0.627482\pi\)
−0.389877 + 0.920867i \(0.627482\pi\)
\(108\) 0 0
\(109\) 7.34155 0.703193 0.351597 0.936152i \(-0.385639\pi\)
0.351597 + 0.936152i \(0.385639\pi\)
\(110\) 0 0
\(111\) 10.0885 0.957562
\(112\) 0 0
\(113\) −0.260722 −0.0245267 −0.0122633 0.999925i \(-0.503904\pi\)
−0.0122633 + 0.999925i \(0.503904\pi\)
\(114\) 0 0
\(115\) 8.19819 0.764485
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11.3836 −1.04353
\(120\) 0 0
\(121\) 21.8750 1.98864
\(122\) 0 0
\(123\) 10.0894 0.909734
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 12.6915 1.12619 0.563096 0.826392i \(-0.309610\pi\)
0.563096 + 0.826392i \(0.309610\pi\)
\(128\) 0 0
\(129\) −10.9435 −0.963522
\(130\) 0 0
\(131\) 0.587367 0.0513185 0.0256592 0.999671i \(-0.491832\pi\)
0.0256592 + 0.999671i \(0.491832\pi\)
\(132\) 0 0
\(133\) 15.2110 1.31896
\(134\) 0 0
\(135\) −5.65673 −0.486854
\(136\) 0 0
\(137\) −15.5251 −1.32640 −0.663200 0.748443i \(-0.730802\pi\)
−0.663200 + 0.748443i \(0.730802\pi\)
\(138\) 0 0
\(139\) 18.8204 1.59632 0.798161 0.602444i \(-0.205806\pi\)
0.798161 + 0.602444i \(0.205806\pi\)
\(140\) 0 0
\(141\) −7.63506 −0.642988
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −9.54822 −0.792936
\(146\) 0 0
\(147\) 3.22467 0.265966
\(148\) 0 0
\(149\) 16.9576 1.38922 0.694609 0.719387i \(-0.255577\pi\)
0.694609 + 0.719387i \(0.255577\pi\)
\(150\) 0 0
\(151\) 10.9444 0.890640 0.445320 0.895371i \(-0.353090\pi\)
0.445320 + 0.895371i \(0.353090\pi\)
\(152\) 0 0
\(153\) 5.32440 0.430453
\(154\) 0 0
\(155\) −2.41917 −0.194313
\(156\) 0 0
\(157\) 10.0706 0.803718 0.401859 0.915702i \(-0.368364\pi\)
0.401859 + 0.915702i \(0.368364\pi\)
\(158\) 0 0
\(159\) −5.55394 −0.440456
\(160\) 0 0
\(161\) −17.7942 −1.40238
\(162\) 0 0
\(163\) 19.5144 1.52849 0.764243 0.644928i \(-0.223113\pi\)
0.764243 + 0.644928i \(0.223113\pi\)
\(164\) 0 0
\(165\) 8.07777 0.628853
\(166\) 0 0
\(167\) 13.4061 1.03740 0.518698 0.854958i \(-0.326417\pi\)
0.518698 + 0.854958i \(0.326417\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −7.11457 −0.544064
\(172\) 0 0
\(173\) 15.1841 1.15443 0.577214 0.816593i \(-0.304140\pi\)
0.577214 + 0.816593i \(0.304140\pi\)
\(174\) 0 0
\(175\) 2.17051 0.164075
\(176\) 0 0
\(177\) −1.40036 −0.105257
\(178\) 0 0
\(179\) 1.24513 0.0930655 0.0465327 0.998917i \(-0.485183\pi\)
0.0465327 + 0.998917i \(0.485183\pi\)
\(180\) 0 0
\(181\) −4.69219 −0.348768 −0.174384 0.984678i \(-0.555793\pi\)
−0.174384 + 0.984678i \(0.555793\pi\)
\(182\) 0 0
\(183\) 6.03968 0.446466
\(184\) 0 0
\(185\) 7.16094 0.526483
\(186\) 0 0
\(187\) −30.0713 −2.19903
\(188\) 0 0
\(189\) 12.2780 0.893091
\(190\) 0 0
\(191\) 11.1007 0.803218 0.401609 0.915811i \(-0.368451\pi\)
0.401609 + 0.915811i \(0.368451\pi\)
\(192\) 0 0
\(193\) 6.30549 0.453879 0.226940 0.973909i \(-0.427128\pi\)
0.226940 + 0.973909i \(0.427128\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.9703 1.13783 0.568917 0.822395i \(-0.307363\pi\)
0.568917 + 0.822395i \(0.307363\pi\)
\(198\) 0 0
\(199\) −12.0075 −0.851191 −0.425596 0.904914i \(-0.639935\pi\)
−0.425596 + 0.904914i \(0.639935\pi\)
\(200\) 0 0
\(201\) 16.3878 1.15590
\(202\) 0 0
\(203\) 20.7245 1.45457
\(204\) 0 0
\(205\) 7.16158 0.500186
\(206\) 0 0
\(207\) 8.32281 0.578475
\(208\) 0 0
\(209\) 40.1818 2.77943
\(210\) 0 0
\(211\) −1.02582 −0.0706203 −0.0353102 0.999376i \(-0.511242\pi\)
−0.0353102 + 0.999376i \(0.511242\pi\)
\(212\) 0 0
\(213\) 7.31285 0.501068
\(214\) 0 0
\(215\) −7.76780 −0.529760
\(216\) 0 0
\(217\) 5.25083 0.356450
\(218\) 0 0
\(219\) −4.31219 −0.291391
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.17108 −0.145387 −0.0726933 0.997354i \(-0.523159\pi\)
−0.0726933 + 0.997354i \(0.523159\pi\)
\(224\) 0 0
\(225\) −1.01520 −0.0676800
\(226\) 0 0
\(227\) −6.48039 −0.430119 −0.215059 0.976601i \(-0.568994\pi\)
−0.215059 + 0.976601i \(0.568994\pi\)
\(228\) 0 0
\(229\) 25.8710 1.70960 0.854801 0.518955i \(-0.173679\pi\)
0.854801 + 0.518955i \(0.173679\pi\)
\(230\) 0 0
\(231\) −17.5328 −1.15358
\(232\) 0 0
\(233\) 26.4139 1.73043 0.865216 0.501400i \(-0.167181\pi\)
0.865216 + 0.501400i \(0.167181\pi\)
\(234\) 0 0
\(235\) −5.41944 −0.353525
\(236\) 0 0
\(237\) −10.3286 −0.670913
\(238\) 0 0
\(239\) 0.157397 0.0101811 0.00509057 0.999987i \(-0.498380\pi\)
0.00509057 + 0.999987i \(0.498380\pi\)
\(240\) 0 0
\(241\) −11.5671 −0.745103 −0.372551 0.928012i \(-0.621517\pi\)
−0.372551 + 0.928012i \(0.621517\pi\)
\(242\) 0 0
\(243\) −10.0334 −0.643646
\(244\) 0 0
\(245\) 2.28890 0.146232
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 4.03131 0.255474
\(250\) 0 0
\(251\) −1.54408 −0.0974615 −0.0487307 0.998812i \(-0.515518\pi\)
−0.0487307 + 0.998812i \(0.515518\pi\)
\(252\) 0 0
\(253\) −47.0057 −2.95523
\(254\) 0 0
\(255\) −7.38886 −0.462709
\(256\) 0 0
\(257\) 14.9633 0.933388 0.466694 0.884419i \(-0.345445\pi\)
0.466694 + 0.884419i \(0.345445\pi\)
\(258\) 0 0
\(259\) −15.5429 −0.965787
\(260\) 0 0
\(261\) −9.69335 −0.600003
\(262\) 0 0
\(263\) −15.1383 −0.933470 −0.466735 0.884397i \(-0.654570\pi\)
−0.466735 + 0.884397i \(0.654570\pi\)
\(264\) 0 0
\(265\) −3.94223 −0.242170
\(266\) 0 0
\(267\) −17.3015 −1.05884
\(268\) 0 0
\(269\) −9.14065 −0.557315 −0.278658 0.960390i \(-0.589889\pi\)
−0.278658 + 0.960390i \(0.589889\pi\)
\(270\) 0 0
\(271\) 4.39788 0.267152 0.133576 0.991039i \(-0.457354\pi\)
0.133576 + 0.991039i \(0.457354\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.73367 0.345753
\(276\) 0 0
\(277\) 26.9216 1.61756 0.808780 0.588112i \(-0.200128\pi\)
0.808780 + 0.588112i \(0.200128\pi\)
\(278\) 0 0
\(279\) −2.45594 −0.147034
\(280\) 0 0
\(281\) −7.40886 −0.441975 −0.220988 0.975277i \(-0.570928\pi\)
−0.220988 + 0.975277i \(0.570928\pi\)
\(282\) 0 0
\(283\) −31.5220 −1.87379 −0.936894 0.349613i \(-0.886313\pi\)
−0.936894 + 0.349613i \(0.886313\pi\)
\(284\) 0 0
\(285\) 9.87313 0.584834
\(286\) 0 0
\(287\) −15.5443 −0.917549
\(288\) 0 0
\(289\) 10.5067 0.618042
\(290\) 0 0
\(291\) −10.3590 −0.607258
\(292\) 0 0
\(293\) −9.19191 −0.536997 −0.268499 0.963280i \(-0.586527\pi\)
−0.268499 + 0.963280i \(0.586527\pi\)
\(294\) 0 0
\(295\) −0.993988 −0.0578722
\(296\) 0 0
\(297\) 32.4338 1.88200
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 16.8601 0.971798
\(302\) 0 0
\(303\) 18.9356 1.08782
\(304\) 0 0
\(305\) 4.28702 0.245474
\(306\) 0 0
\(307\) −20.6454 −1.17830 −0.589148 0.808025i \(-0.700537\pi\)
−0.589148 + 0.808025i \(0.700537\pi\)
\(308\) 0 0
\(309\) −10.8209 −0.615577
\(310\) 0 0
\(311\) 8.12701 0.460841 0.230420 0.973091i \(-0.425990\pi\)
0.230420 + 0.973091i \(0.425990\pi\)
\(312\) 0 0
\(313\) −5.45768 −0.308486 −0.154243 0.988033i \(-0.549294\pi\)
−0.154243 + 0.988033i \(0.549294\pi\)
\(314\) 0 0
\(315\) 2.20350 0.124153
\(316\) 0 0
\(317\) 10.7778 0.605342 0.302671 0.953095i \(-0.402122\pi\)
0.302671 + 0.953095i \(0.402122\pi\)
\(318\) 0 0
\(319\) 54.7464 3.06521
\(320\) 0 0
\(321\) 11.3634 0.634242
\(322\) 0 0
\(323\) −36.7550 −2.04510
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −10.3430 −0.571969
\(328\) 0 0
\(329\) 11.7629 0.648511
\(330\) 0 0
\(331\) −25.9764 −1.42779 −0.713896 0.700252i \(-0.753071\pi\)
−0.713896 + 0.700252i \(0.753071\pi\)
\(332\) 0 0
\(333\) 7.26979 0.398382
\(334\) 0 0
\(335\) 11.6322 0.635534
\(336\) 0 0
\(337\) 20.6275 1.12365 0.561824 0.827256i \(-0.310100\pi\)
0.561824 + 0.827256i \(0.310100\pi\)
\(338\) 0 0
\(339\) 0.367313 0.0199497
\(340\) 0 0
\(341\) 13.8707 0.751143
\(342\) 0 0
\(343\) −20.1616 −1.08863
\(344\) 0 0
\(345\) −11.5499 −0.621823
\(346\) 0 0
\(347\) 18.9341 1.01643 0.508217 0.861229i \(-0.330305\pi\)
0.508217 + 0.861229i \(0.330305\pi\)
\(348\) 0 0
\(349\) 4.53125 0.242552 0.121276 0.992619i \(-0.461301\pi\)
0.121276 + 0.992619i \(0.461301\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.8925 −1.00554 −0.502772 0.864419i \(-0.667686\pi\)
−0.502772 + 0.864419i \(0.667686\pi\)
\(354\) 0 0
\(355\) 5.19073 0.275495
\(356\) 0 0
\(357\) 16.0376 0.848799
\(358\) 0 0
\(359\) 20.4677 1.08024 0.540122 0.841587i \(-0.318378\pi\)
0.540122 + 0.841587i \(0.318378\pi\)
\(360\) 0 0
\(361\) 30.1127 1.58488
\(362\) 0 0
\(363\) −30.8181 −1.61753
\(364\) 0 0
\(365\) −3.06083 −0.160211
\(366\) 0 0
\(367\) 22.6792 1.18384 0.591922 0.805995i \(-0.298369\pi\)
0.591922 + 0.805995i \(0.298369\pi\)
\(368\) 0 0
\(369\) 7.27044 0.378484
\(370\) 0 0
\(371\) 8.55665 0.444239
\(372\) 0 0
\(373\) 14.9049 0.771749 0.385874 0.922551i \(-0.373900\pi\)
0.385874 + 0.922551i \(0.373900\pi\)
\(374\) 0 0
\(375\) 1.40883 0.0727516
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 15.7604 0.809555 0.404778 0.914415i \(-0.367349\pi\)
0.404778 + 0.914415i \(0.367349\pi\)
\(380\) 0 0
\(381\) −17.8802 −0.916030
\(382\) 0 0
\(383\) 1.21077 0.0618675 0.0309338 0.999521i \(-0.490152\pi\)
0.0309338 + 0.999521i \(0.490152\pi\)
\(384\) 0 0
\(385\) −12.4450 −0.634255
\(386\) 0 0
\(387\) −7.88588 −0.400862
\(388\) 0 0
\(389\) −30.4592 −1.54434 −0.772172 0.635414i \(-0.780829\pi\)
−0.772172 + 0.635414i \(0.780829\pi\)
\(390\) 0 0
\(391\) 42.9969 2.17445
\(392\) 0 0
\(393\) −0.827500 −0.0417418
\(394\) 0 0
\(395\) −7.33131 −0.368878
\(396\) 0 0
\(397\) −30.6666 −1.53911 −0.769557 0.638578i \(-0.779523\pi\)
−0.769557 + 0.638578i \(0.779523\pi\)
\(398\) 0 0
\(399\) −21.4297 −1.07283
\(400\) 0 0
\(401\) −12.3885 −0.618652 −0.309326 0.950956i \(-0.600104\pi\)
−0.309326 + 0.950956i \(0.600104\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.92377 0.244664
\(406\) 0 0
\(407\) −41.0585 −2.03519
\(408\) 0 0
\(409\) 9.76935 0.483063 0.241532 0.970393i \(-0.422350\pi\)
0.241532 + 0.970393i \(0.422350\pi\)
\(410\) 0 0
\(411\) 21.8722 1.07888
\(412\) 0 0
\(413\) 2.15746 0.106162
\(414\) 0 0
\(415\) 2.86146 0.140464
\(416\) 0 0
\(417\) −26.5147 −1.29843
\(418\) 0 0
\(419\) 5.65383 0.276208 0.138104 0.990418i \(-0.455899\pi\)
0.138104 + 0.990418i \(0.455899\pi\)
\(420\) 0 0
\(421\) 17.8610 0.870494 0.435247 0.900311i \(-0.356661\pi\)
0.435247 + 0.900311i \(0.356661\pi\)
\(422\) 0 0
\(423\) −5.50182 −0.267507
\(424\) 0 0
\(425\) −5.24468 −0.254405
\(426\) 0 0
\(427\) −9.30501 −0.450301
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −27.3930 −1.31947 −0.659737 0.751497i \(-0.729332\pi\)
−0.659737 + 0.751497i \(0.729332\pi\)
\(432\) 0 0
\(433\) −10.0057 −0.480841 −0.240421 0.970669i \(-0.577285\pi\)
−0.240421 + 0.970669i \(0.577285\pi\)
\(434\) 0 0
\(435\) 13.4518 0.644965
\(436\) 0 0
\(437\) −57.4533 −2.74836
\(438\) 0 0
\(439\) −1.38320 −0.0660167 −0.0330084 0.999455i \(-0.510509\pi\)
−0.0330084 + 0.999455i \(0.510509\pi\)
\(440\) 0 0
\(441\) 2.32369 0.110652
\(442\) 0 0
\(443\) 15.6990 0.745883 0.372941 0.927855i \(-0.378349\pi\)
0.372941 + 0.927855i \(0.378349\pi\)
\(444\) 0 0
\(445\) −12.2808 −0.582166
\(446\) 0 0
\(447\) −23.8903 −1.12997
\(448\) 0 0
\(449\) 29.7220 1.40267 0.701335 0.712832i \(-0.252588\pi\)
0.701335 + 0.712832i \(0.252588\pi\)
\(450\) 0 0
\(451\) −41.0622 −1.93354
\(452\) 0 0
\(453\) −15.4187 −0.724436
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.208226 −0.00974041 −0.00487021 0.999988i \(-0.501550\pi\)
−0.00487021 + 0.999988i \(0.501550\pi\)
\(458\) 0 0
\(459\) −29.6678 −1.38477
\(460\) 0 0
\(461\) 6.24475 0.290847 0.145424 0.989369i \(-0.453546\pi\)
0.145424 + 0.989369i \(0.453546\pi\)
\(462\) 0 0
\(463\) −3.42664 −0.159249 −0.0796246 0.996825i \(-0.525372\pi\)
−0.0796246 + 0.996825i \(0.525372\pi\)
\(464\) 0 0
\(465\) 3.40820 0.158052
\(466\) 0 0
\(467\) 26.2665 1.21547 0.607733 0.794141i \(-0.292079\pi\)
0.607733 + 0.794141i \(0.292079\pi\)
\(468\) 0 0
\(469\) −25.2477 −1.16583
\(470\) 0 0
\(471\) −14.1877 −0.653735
\(472\) 0 0
\(473\) 44.5380 2.04786
\(474\) 0 0
\(475\) 7.00804 0.321551
\(476\) 0 0
\(477\) −4.00216 −0.183246
\(478\) 0 0
\(479\) −3.46962 −0.158531 −0.0792654 0.996854i \(-0.525257\pi\)
−0.0792654 + 0.996854i \(0.525257\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 25.0690 1.14068
\(484\) 0 0
\(485\) −7.35295 −0.333880
\(486\) 0 0
\(487\) −16.6306 −0.753606 −0.376803 0.926293i \(-0.622977\pi\)
−0.376803 + 0.926293i \(0.622977\pi\)
\(488\) 0 0
\(489\) −27.4925 −1.24325
\(490\) 0 0
\(491\) 19.0416 0.859334 0.429667 0.902987i \(-0.358631\pi\)
0.429667 + 0.902987i \(0.358631\pi\)
\(492\) 0 0
\(493\) −50.0774 −2.25537
\(494\) 0 0
\(495\) 5.82083 0.261627
\(496\) 0 0
\(497\) −11.2665 −0.505372
\(498\) 0 0
\(499\) 34.4954 1.54423 0.772114 0.635485i \(-0.219200\pi\)
0.772114 + 0.635485i \(0.219200\pi\)
\(500\) 0 0
\(501\) −18.8869 −0.843805
\(502\) 0 0
\(503\) −23.5640 −1.05067 −0.525333 0.850897i \(-0.676059\pi\)
−0.525333 + 0.850897i \(0.676059\pi\)
\(504\) 0 0
\(505\) 13.4407 0.598102
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −25.8268 −1.14475 −0.572377 0.819991i \(-0.693978\pi\)
−0.572377 + 0.819991i \(0.693978\pi\)
\(510\) 0 0
\(511\) 6.64355 0.293893
\(512\) 0 0
\(513\) 39.6426 1.75026
\(514\) 0 0
\(515\) −7.68074 −0.338454
\(516\) 0 0
\(517\) 31.0733 1.36660
\(518\) 0 0
\(519\) −21.3918 −0.938998
\(520\) 0 0
\(521\) −24.7194 −1.08298 −0.541489 0.840708i \(-0.682139\pi\)
−0.541489 + 0.840708i \(0.682139\pi\)
\(522\) 0 0
\(523\) −22.6317 −0.989613 −0.494807 0.869003i \(-0.664761\pi\)
−0.494807 + 0.869003i \(0.664761\pi\)
\(524\) 0 0
\(525\) −3.05787 −0.133457
\(526\) 0 0
\(527\) −12.6878 −0.552689
\(528\) 0 0
\(529\) 44.2104 1.92219
\(530\) 0 0
\(531\) −1.00910 −0.0437911
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 8.06583 0.348716
\(536\) 0 0
\(537\) −1.75418 −0.0756983
\(538\) 0 0
\(539\) −13.1238 −0.565282
\(540\) 0 0
\(541\) 16.2447 0.698413 0.349206 0.937046i \(-0.386451\pi\)
0.349206 + 0.937046i \(0.386451\pi\)
\(542\) 0 0
\(543\) 6.61049 0.283683
\(544\) 0 0
\(545\) −7.34155 −0.314478
\(546\) 0 0
\(547\) −24.2073 −1.03503 −0.517515 0.855674i \(-0.673143\pi\)
−0.517515 + 0.855674i \(0.673143\pi\)
\(548\) 0 0
\(549\) 4.35219 0.185747
\(550\) 0 0
\(551\) 66.9143 2.85065
\(552\) 0 0
\(553\) 15.9127 0.676675
\(554\) 0 0
\(555\) −10.0885 −0.428235
\(556\) 0 0
\(557\) 16.6266 0.704492 0.352246 0.935907i \(-0.385418\pi\)
0.352246 + 0.935907i \(0.385418\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 42.3653 1.78867
\(562\) 0 0
\(563\) −2.94274 −0.124022 −0.0620109 0.998075i \(-0.519751\pi\)
−0.0620109 + 0.998075i \(0.519751\pi\)
\(564\) 0 0
\(565\) 0.260722 0.0109687
\(566\) 0 0
\(567\) −10.6871 −0.448815
\(568\) 0 0
\(569\) −45.8747 −1.92317 −0.961583 0.274514i \(-0.911483\pi\)
−0.961583 + 0.274514i \(0.911483\pi\)
\(570\) 0 0
\(571\) −2.07821 −0.0869705 −0.0434853 0.999054i \(-0.513846\pi\)
−0.0434853 + 0.999054i \(0.513846\pi\)
\(572\) 0 0
\(573\) −15.6390 −0.653328
\(574\) 0 0
\(575\) −8.19819 −0.341888
\(576\) 0 0
\(577\) −42.5167 −1.76999 −0.884997 0.465597i \(-0.845839\pi\)
−0.884997 + 0.465597i \(0.845839\pi\)
\(578\) 0 0
\(579\) −8.88336 −0.369180
\(580\) 0 0
\(581\) −6.21082 −0.257668
\(582\) 0 0
\(583\) 22.6035 0.936141
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.29519 −0.342379 −0.171190 0.985238i \(-0.554761\pi\)
−0.171190 + 0.985238i \(0.554761\pi\)
\(588\) 0 0
\(589\) 16.9537 0.698563
\(590\) 0 0
\(591\) −22.4994 −0.925500
\(592\) 0 0
\(593\) −20.8549 −0.856407 −0.428203 0.903682i \(-0.640853\pi\)
−0.428203 + 0.903682i \(0.640853\pi\)
\(594\) 0 0
\(595\) 11.3836 0.466683
\(596\) 0 0
\(597\) 16.9166 0.692349
\(598\) 0 0
\(599\) −19.1748 −0.783461 −0.391730 0.920080i \(-0.628123\pi\)
−0.391730 + 0.920080i \(0.628123\pi\)
\(600\) 0 0
\(601\) 46.9333 1.91445 0.957225 0.289345i \(-0.0934375\pi\)
0.957225 + 0.289345i \(0.0934375\pi\)
\(602\) 0 0
\(603\) 11.8090 0.480899
\(604\) 0 0
\(605\) −21.8750 −0.889345
\(606\) 0 0
\(607\) 7.58350 0.307805 0.153902 0.988086i \(-0.450816\pi\)
0.153902 + 0.988086i \(0.450816\pi\)
\(608\) 0 0
\(609\) −29.1972 −1.18313
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 14.1514 0.571569 0.285785 0.958294i \(-0.407746\pi\)
0.285785 + 0.958294i \(0.407746\pi\)
\(614\) 0 0
\(615\) −10.0894 −0.406846
\(616\) 0 0
\(617\) −13.2765 −0.534493 −0.267246 0.963628i \(-0.586114\pi\)
−0.267246 + 0.963628i \(0.586114\pi\)
\(618\) 0 0
\(619\) 28.0262 1.12647 0.563234 0.826298i \(-0.309557\pi\)
0.563234 + 0.826298i \(0.309557\pi\)
\(620\) 0 0
\(621\) −46.3750 −1.86096
\(622\) 0 0
\(623\) 26.6556 1.06793
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −56.6093 −2.26076
\(628\) 0 0
\(629\) 37.5569 1.49749
\(630\) 0 0
\(631\) −18.1132 −0.721074 −0.360537 0.932745i \(-0.617406\pi\)
−0.360537 + 0.932745i \(0.617406\pi\)
\(632\) 0 0
\(633\) 1.44520 0.0574417
\(634\) 0 0
\(635\) −12.6915 −0.503648
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 5.26963 0.208463
\(640\) 0 0
\(641\) 11.1409 0.440038 0.220019 0.975496i \(-0.429388\pi\)
0.220019 + 0.975496i \(0.429388\pi\)
\(642\) 0 0
\(643\) −4.70617 −0.185593 −0.0927965 0.995685i \(-0.529581\pi\)
−0.0927965 + 0.995685i \(0.529581\pi\)
\(644\) 0 0
\(645\) 10.9435 0.430900
\(646\) 0 0
\(647\) 6.21648 0.244395 0.122197 0.992506i \(-0.461006\pi\)
0.122197 + 0.992506i \(0.461006\pi\)
\(648\) 0 0
\(649\) 5.69920 0.223713
\(650\) 0 0
\(651\) −7.39752 −0.289932
\(652\) 0 0
\(653\) 33.0832 1.29465 0.647323 0.762215i \(-0.275888\pi\)
0.647323 + 0.762215i \(0.275888\pi\)
\(654\) 0 0
\(655\) −0.587367 −0.0229503
\(656\) 0 0
\(657\) −3.10735 −0.121229
\(658\) 0 0
\(659\) −10.1051 −0.393640 −0.196820 0.980440i \(-0.563061\pi\)
−0.196820 + 0.980440i \(0.563061\pi\)
\(660\) 0 0
\(661\) −28.5808 −1.11166 −0.555832 0.831295i \(-0.687600\pi\)
−0.555832 + 0.831295i \(0.687600\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −15.2110 −0.589857
\(666\) 0 0
\(667\) −78.2781 −3.03094
\(668\) 0 0
\(669\) 3.05869 0.118256
\(670\) 0 0
\(671\) −24.5804 −0.948915
\(672\) 0 0
\(673\) −27.0625 −1.04318 −0.521590 0.853196i \(-0.674661\pi\)
−0.521590 + 0.853196i \(0.674661\pi\)
\(674\) 0 0
\(675\) 5.65673 0.217728
\(676\) 0 0
\(677\) −48.4114 −1.86060 −0.930301 0.366798i \(-0.880454\pi\)
−0.930301 + 0.366798i \(0.880454\pi\)
\(678\) 0 0
\(679\) 15.9596 0.612474
\(680\) 0 0
\(681\) 9.12976 0.349853
\(682\) 0 0
\(683\) 20.6854 0.791507 0.395753 0.918357i \(-0.370484\pi\)
0.395753 + 0.918357i \(0.370484\pi\)
\(684\) 0 0
\(685\) 15.5251 0.593184
\(686\) 0 0
\(687\) −36.4478 −1.39057
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 28.3098 1.07696 0.538478 0.842639i \(-0.318999\pi\)
0.538478 + 0.842639i \(0.318999\pi\)
\(692\) 0 0
\(693\) −12.6341 −0.479931
\(694\) 0 0
\(695\) −18.8204 −0.713897
\(696\) 0 0
\(697\) 37.5602 1.42269
\(698\) 0 0
\(699\) −37.2127 −1.40751
\(700\) 0 0
\(701\) −12.7481 −0.481490 −0.240745 0.970588i \(-0.577392\pi\)
−0.240745 + 0.970588i \(0.577392\pi\)
\(702\) 0 0
\(703\) −50.1842 −1.89273
\(704\) 0 0
\(705\) 7.63506 0.287553
\(706\) 0 0
\(707\) −29.1731 −1.09717
\(708\) 0 0
\(709\) −44.3115 −1.66415 −0.832077 0.554660i \(-0.812848\pi\)
−0.832077 + 0.554660i \(0.812848\pi\)
\(710\) 0 0
\(711\) −7.44275 −0.279125
\(712\) 0 0
\(713\) −19.8328 −0.742746
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −0.221745 −0.00828122
\(718\) 0 0
\(719\) 7.49703 0.279592 0.139796 0.990180i \(-0.455355\pi\)
0.139796 + 0.990180i \(0.455355\pi\)
\(720\) 0 0
\(721\) 16.6711 0.620864
\(722\) 0 0
\(723\) 16.2961 0.606058
\(724\) 0 0
\(725\) 9.54822 0.354612
\(726\) 0 0
\(727\) 23.3642 0.866530 0.433265 0.901267i \(-0.357361\pi\)
0.433265 + 0.901267i \(0.357361\pi\)
\(728\) 0 0
\(729\) 28.9067 1.07062
\(730\) 0 0
\(731\) −40.7397 −1.50681
\(732\) 0 0
\(733\) 27.2189 1.00535 0.502677 0.864474i \(-0.332348\pi\)
0.502677 + 0.864474i \(0.332348\pi\)
\(734\) 0 0
\(735\) −3.22467 −0.118944
\(736\) 0 0
\(737\) −66.6951 −2.45675
\(738\) 0 0
\(739\) −35.2389 −1.29628 −0.648141 0.761520i \(-0.724453\pi\)
−0.648141 + 0.761520i \(0.724453\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.76023 0.358068 0.179034 0.983843i \(-0.442703\pi\)
0.179034 + 0.983843i \(0.442703\pi\)
\(744\) 0 0
\(745\) −16.9576 −0.621277
\(746\) 0 0
\(747\) 2.90496 0.106287
\(748\) 0 0
\(749\) −17.5069 −0.639690
\(750\) 0 0
\(751\) 53.1333 1.93886 0.969431 0.245363i \(-0.0789073\pi\)
0.969431 + 0.245363i \(0.0789073\pi\)
\(752\) 0 0
\(753\) 2.17534 0.0792740
\(754\) 0 0
\(755\) −10.9444 −0.398306
\(756\) 0 0
\(757\) −33.6611 −1.22343 −0.611716 0.791077i \(-0.709520\pi\)
−0.611716 + 0.791077i \(0.709520\pi\)
\(758\) 0 0
\(759\) 66.2231 2.40375
\(760\) 0 0
\(761\) 38.2985 1.38832 0.694160 0.719821i \(-0.255776\pi\)
0.694160 + 0.719821i \(0.255776\pi\)
\(762\) 0 0
\(763\) 15.9349 0.576882
\(764\) 0 0
\(765\) −5.32440 −0.192504
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 28.9380 1.04353 0.521765 0.853089i \(-0.325274\pi\)
0.521765 + 0.853089i \(0.325274\pi\)
\(770\) 0 0
\(771\) −21.0808 −0.759206
\(772\) 0 0
\(773\) −22.4859 −0.808762 −0.404381 0.914591i \(-0.632513\pi\)
−0.404381 + 0.914591i \(0.632513\pi\)
\(774\) 0 0
\(775\) 2.41917 0.0868992
\(776\) 0 0
\(777\) 21.8972 0.785559
\(778\) 0 0
\(779\) −50.1887 −1.79820
\(780\) 0 0
\(781\) −29.7619 −1.06497
\(782\) 0 0
\(783\) 54.0117 1.93022
\(784\) 0 0
\(785\) −10.0706 −0.359434
\(786\) 0 0
\(787\) −16.0779 −0.573117 −0.286558 0.958063i \(-0.592511\pi\)
−0.286558 + 0.958063i \(0.592511\pi\)
\(788\) 0 0
\(789\) 21.3273 0.759273
\(790\) 0 0
\(791\) −0.565899 −0.0201211
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 5.55394 0.196978
\(796\) 0 0
\(797\) −51.8604 −1.83699 −0.918494 0.395434i \(-0.870594\pi\)
−0.918494 + 0.395434i \(0.870594\pi\)
\(798\) 0 0
\(799\) −28.4232 −1.00554
\(800\) 0 0
\(801\) −12.4675 −0.440516
\(802\) 0 0
\(803\) 17.5498 0.619319
\(804\) 0 0
\(805\) 17.7942 0.627164
\(806\) 0 0
\(807\) 12.8776 0.453314
\(808\) 0 0
\(809\) 26.9372 0.947060 0.473530 0.880778i \(-0.342980\pi\)
0.473530 + 0.880778i \(0.342980\pi\)
\(810\) 0 0
\(811\) −36.5575 −1.28371 −0.641854 0.766827i \(-0.721834\pi\)
−0.641854 + 0.766827i \(0.721834\pi\)
\(812\) 0 0
\(813\) −6.19586 −0.217298
\(814\) 0 0
\(815\) −19.5144 −0.683560
\(816\) 0 0
\(817\) 54.4371 1.90451
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.59771 −0.0906607 −0.0453304 0.998972i \(-0.514434\pi\)
−0.0453304 + 0.998972i \(0.514434\pi\)
\(822\) 0 0
\(823\) −7.85046 −0.273650 −0.136825 0.990595i \(-0.543690\pi\)
−0.136825 + 0.990595i \(0.543690\pi\)
\(824\) 0 0
\(825\) −8.07777 −0.281232
\(826\) 0 0
\(827\) −7.12878 −0.247892 −0.123946 0.992289i \(-0.539555\pi\)
−0.123946 + 0.992289i \(0.539555\pi\)
\(828\) 0 0
\(829\) 51.3687 1.78411 0.892054 0.451929i \(-0.149264\pi\)
0.892054 + 0.451929i \(0.149264\pi\)
\(830\) 0 0
\(831\) −37.9279 −1.31570
\(832\) 0 0
\(833\) 12.0046 0.415933
\(834\) 0 0
\(835\) −13.4061 −0.463938
\(836\) 0 0
\(837\) 13.6846 0.473009
\(838\) 0 0
\(839\) −0.322289 −0.0111266 −0.00556332 0.999985i \(-0.501771\pi\)
−0.00556332 + 0.999985i \(0.501771\pi\)
\(840\) 0 0
\(841\) 62.1685 2.14374
\(842\) 0 0
\(843\) 10.4378 0.359497
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 47.4798 1.63143
\(848\) 0 0
\(849\) 44.4091 1.52412
\(850\) 0 0
\(851\) 58.7068 2.01244
\(852\) 0 0
\(853\) −18.1009 −0.619763 −0.309881 0.950775i \(-0.600289\pi\)
−0.309881 + 0.950775i \(0.600289\pi\)
\(854\) 0 0
\(855\) 7.11457 0.243313
\(856\) 0 0
\(857\) −44.3079 −1.51353 −0.756764 0.653688i \(-0.773221\pi\)
−0.756764 + 0.653688i \(0.773221\pi\)
\(858\) 0 0
\(859\) 14.5314 0.495807 0.247903 0.968785i \(-0.420258\pi\)
0.247903 + 0.968785i \(0.420258\pi\)
\(860\) 0 0
\(861\) 21.8992 0.746323
\(862\) 0 0
\(863\) 23.5161 0.800496 0.400248 0.916407i \(-0.368924\pi\)
0.400248 + 0.916407i \(0.368924\pi\)
\(864\) 0 0
\(865\) −15.1841 −0.516276
\(866\) 0 0
\(867\) −14.8022 −0.502707
\(868\) 0 0
\(869\) 42.0354 1.42595
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −7.46471 −0.252642
\(874\) 0 0
\(875\) −2.17051 −0.0733765
\(876\) 0 0
\(877\) −27.6427 −0.933428 −0.466714 0.884408i \(-0.654562\pi\)
−0.466714 + 0.884408i \(0.654562\pi\)
\(878\) 0 0
\(879\) 12.9498 0.436787
\(880\) 0 0
\(881\) −11.7134 −0.394635 −0.197318 0.980340i \(-0.563223\pi\)
−0.197318 + 0.980340i \(0.563223\pi\)
\(882\) 0 0
\(883\) −5.53015 −0.186105 −0.0930523 0.995661i \(-0.529662\pi\)
−0.0930523 + 0.995661i \(0.529662\pi\)
\(884\) 0 0
\(885\) 1.40036 0.0470726
\(886\) 0 0
\(887\) −15.0601 −0.505668 −0.252834 0.967510i \(-0.581363\pi\)
−0.252834 + 0.967510i \(0.581363\pi\)
\(888\) 0 0
\(889\) 27.5471 0.923899
\(890\) 0 0
\(891\) −28.2313 −0.945783
\(892\) 0 0
\(893\) 37.9797 1.27094
\(894\) 0 0
\(895\) −1.24513 −0.0416202
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 23.0988 0.770388
\(900\) 0 0
\(901\) −20.6758 −0.688810
\(902\) 0 0
\(903\) −23.7530 −0.790449
\(904\) 0 0
\(905\) 4.69219 0.155974
\(906\) 0 0
\(907\) 47.0337 1.56173 0.780865 0.624700i \(-0.214779\pi\)
0.780865 + 0.624700i \(0.214779\pi\)
\(908\) 0 0
\(909\) 13.6450 0.452575
\(910\) 0 0
\(911\) 18.2632 0.605086 0.302543 0.953136i \(-0.402164\pi\)
0.302543 + 0.953136i \(0.402164\pi\)
\(912\) 0 0
\(913\) −16.4067 −0.542982
\(914\) 0 0
\(915\) −6.03968 −0.199666
\(916\) 0 0
\(917\) 1.27488 0.0421004
\(918\) 0 0
\(919\) 23.9725 0.790780 0.395390 0.918513i \(-0.370610\pi\)
0.395390 + 0.918513i \(0.370610\pi\)
\(920\) 0 0
\(921\) 29.0859 0.958412
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −7.16094 −0.235450
\(926\) 0 0
\(927\) −7.79749 −0.256103
\(928\) 0 0
\(929\) −31.2931 −1.02669 −0.513347 0.858181i \(-0.671595\pi\)
−0.513347 + 0.858181i \(0.671595\pi\)
\(930\) 0 0
\(931\) −16.0407 −0.525713
\(932\) 0 0
\(933\) −11.4496 −0.374842
\(934\) 0 0
\(935\) 30.0713 0.983437
\(936\) 0 0
\(937\) −0.100918 −0.00329684 −0.00164842 0.999999i \(-0.500525\pi\)
−0.00164842 + 0.999999i \(0.500525\pi\)
\(938\) 0 0
\(939\) 7.68894 0.250919
\(940\) 0 0
\(941\) −32.1209 −1.04711 −0.523555 0.851992i \(-0.675395\pi\)
−0.523555 + 0.851992i \(0.675395\pi\)
\(942\) 0 0
\(943\) 58.7120 1.91193
\(944\) 0 0
\(945\) −12.2780 −0.399403
\(946\) 0 0
\(947\) −11.3926 −0.370209 −0.185104 0.982719i \(-0.559262\pi\)
−0.185104 + 0.982719i \(0.559262\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −15.1841 −0.492378
\(952\) 0 0
\(953\) 20.0389 0.649125 0.324562 0.945864i \(-0.394783\pi\)
0.324562 + 0.945864i \(0.394783\pi\)
\(954\) 0 0
\(955\) −11.1007 −0.359210
\(956\) 0 0
\(957\) −77.1283 −2.49320
\(958\) 0 0
\(959\) −33.6973 −1.08814
\(960\) 0 0
\(961\) −25.1476 −0.811213
\(962\) 0 0
\(963\) 8.18844 0.263869
\(964\) 0 0
\(965\) −6.30549 −0.202981
\(966\) 0 0
\(967\) −17.7835 −0.571878 −0.285939 0.958248i \(-0.592306\pi\)
−0.285939 + 0.958248i \(0.592306\pi\)
\(968\) 0 0
\(969\) 51.7815 1.66346
\(970\) 0 0
\(971\) 10.0669 0.323062 0.161531 0.986868i \(-0.448357\pi\)
0.161531 + 0.986868i \(0.448357\pi\)
\(972\) 0 0
\(973\) 40.8497 1.30958
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.91072 0.221094 0.110547 0.993871i \(-0.464740\pi\)
0.110547 + 0.993871i \(0.464740\pi\)
\(978\) 0 0
\(979\) 70.4141 2.25044
\(980\) 0 0
\(981\) −7.45315 −0.237961
\(982\) 0 0
\(983\) 29.7862 0.950033 0.475017 0.879977i \(-0.342442\pi\)
0.475017 + 0.879977i \(0.342442\pi\)
\(984\) 0 0
\(985\) −15.9703 −0.508855
\(986\) 0 0
\(987\) −16.5720 −0.527491
\(988\) 0 0
\(989\) −63.6820 −2.02497
\(990\) 0 0
\(991\) −19.6830 −0.625252 −0.312626 0.949876i \(-0.601209\pi\)
−0.312626 + 0.949876i \(0.601209\pi\)
\(992\) 0 0
\(993\) 36.5963 1.16135
\(994\) 0 0
\(995\) 12.0075 0.380664
\(996\) 0 0
\(997\) 19.3853 0.613940 0.306970 0.951719i \(-0.400685\pi\)
0.306970 + 0.951719i \(0.400685\pi\)
\(998\) 0 0
\(999\) −40.5075 −1.28160
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3380.2.a.r.1.3 9
13.5 odd 4 3380.2.f.j.3041.6 18
13.8 odd 4 3380.2.f.j.3041.5 18
13.12 even 2 3380.2.a.s.1.3 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.r.1.3 9 1.1 even 1 trivial
3380.2.a.s.1.3 yes 9 13.12 even 2
3380.2.f.j.3041.5 18 13.8 odd 4
3380.2.f.j.3041.6 18 13.5 odd 4